How Can a Curve Have Infinite Area but Finite Volume?

[drcrabs]

Whence evaluating the area under the curve

$$y=\frac{1}{x} \\\ \mbox{for} \\\ 1 \leq x < \infty$$

it evaluates to $$\infty$$

But when evaluating the volume using

$$Volume = \pi \int y^2 dx \\\ \mbox{on} \\\ a \leq x < b$$

hence


$$Volume = \pi \int \frac{1}{x^2} \\dx$$

hence

$$Volume = \pi [-\ \frac{1}{x}] \\\ \mbox{on} \\\ 1 \leq x < \infty$$

hence

$$Volume = \pi [0 - - 1] = \pi$$

A finite value!

Im having trouble comprehending such concepts and ideas.
Can someone please explain?

 

[Brad Barker]

Whence evaluating the area under the curve

$$y=\frac{1}{x} \\\ \mbox{for} \\\ 1 \leq x < \infty$$

it evaluates to $$\infty$$

But when evaluating the volume using

$$Volume = \pi \int y^2 dx \\\ \mbox{on} \\\ a \leq x < b$$

hence


$$Volume = \pi \int \frac{1}{x^2} \\dx$$

hence

$$Volume = \pi [-\ \frac{1}{x}] \\\ \mbox{on} \\\ 1 \leq x < \infty$$

hence

$$Volume = \pi [0 - - 1] = \pi$$

A finite value!

Im having trouble comprehending such concepts and ideas.
Can someone please explain?

for a more direct "blow your mind" property, the surface area of that solid of revolution is infinite (and the volume, like you said, finite).

things dealing with infinity get pretty strange.

there are fractals that exhibit similar properties, like infinite surface area but zero volume and such.

i deal with it by just casting it off as math.

 

[R0man]

The concept of infinite area but finite volume can be difficult to grasp at first, but it is a result of the properties of the mathematical functions involved. In this case, the function y = 1/x has an infinite area because it continues to increase without bound as x approaches infinity. This means that when we calculate the area under the curve from x=1 to x=infinity, the value will also approach infinity.

However, when we use the same function to calculate the volume of a solid of revolution (like a cone or a cylinder), the result is a finite value. This is because when we integrate y^2, we are essentially taking the cross-sectional area at each point along the x-axis and adding them together. Since the function 1/x decreases as x increases, the cross-sectional area also decreases. This results in a finite volume when we integrate from x=1 to x=infinity.

In simpler terms, the infinite area is a result of the function increasing without bound, while the finite volume is a result of the function decreasing as we move along the x-axis. This is a common occurrence in mathematics and can be seen in other functions as well. It may seem counterintuitive, but it is a result of the properties of the functions involved and can be explained through mathematical reasoning and calculations.